Multinomial Distribution

A generalization of the Binomial distribution for more than two outcomes.

Beyond Success or Failure

The Multinomial distribution extends the Binomial distribution to situations with more than two possible outcomes for each trial. While Binomial models the number of successes in a series of 'success/failure' trials, Multinomial models the number of times each of a set of possible outcomes occurs.

For example, instead of just a 'win' or 'loss', a trade could result in a 'big win', 'small win', 'breakeven', 'small loss', or 'big loss'. The Multinomial distribution can calculate the probability of observing a specific count for each of these categories over a series of trades.

The Formula

The probability of observing a specific set of counts is:

P(X_1=x_1, ..., X_c=x_c) = \frac{n!}{x_1!...x_c!} p_1^{x_1} \cdots p_c^{x_c}

$n$ is the total number of trials.
$c$ is the number of possible outcomes.
$x_i$ is the number of times outcome $i$ occurred.
$p_i$ is the probability of outcome $i$ on a single trial.

Interactive Multinomial Calculator

Specify the parameters to calculate the probability of a specific result.

Total Number of Trials (n): 10

Outcome	Probability ( $p_i$ )	Desired Count ( $x_i$ )
Win
Loss
Draw